# Properties

 Base field $$\Q(\sqrt{19})$$ Label 2.2.76.1-171.1-e1 Conductor $$(3 a)$$ Conductor norm $$171$$ CM no base-change yes: 57.a1,17328.c1 Q-curve yes Torsion order $$1$$ Rank $$2$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field $$\Q(\sqrt{19})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 19$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 19)

gp: K = nfinit(a^2 - 19);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);

## Weierstrass equation

$$y^2+y=x^{3}-x^{2}-2x+2$$
sage: E = EllipticCurve(K, [0, -1, 1, -2, 2])

gp: E = ellinit([0, -1, 1, -2, 2],K)

magma: E := ChangeRing(EllipticCurve([0, -1, 1, -2, 2]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(3 a)$$ = $$\left(-a - 4\right) \cdot \left(-a + 4\right) \cdot \left(a\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$171$$ = $$3^{2} \cdot 19$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(171)$$ = $$\left(-a - 4\right)^{2} \cdot \left(-a + 4\right)^{2} \cdot \left(a\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$29241$$ = $$3^{4} \cdot 19^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{1404928}{171}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$2$$

sage: E.rank()

magma: Rank(E);

Generators: $\left(-\frac{6}{25} a + \frac{53}{25} : \frac{54}{125} a - \frac{352}{125} : 1\right)$, $\left(a + 3 : -2 a - 10 : 1\right)$

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

Heights: 1.67111521371413, 0.464747044349561

sage: [P.height() for P in gens]

magma: [Height(P):P in gens];

Regulator: 0.0142860183735661

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-a - 4\right)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(-a + 4\right)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(a\right)$$ $$19$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ .

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 171.1-e consists of this curve only.

## Base change

This curve is the base-change of elliptic curves 57.a1, 17328.c1, defined over $$\Q$$, so it is also a $$\Q$$-curve.